The $F$-Symbols for Transparent Haagerup-Izumi Categories with $G = \mathbb{Z}_{2n+1}$

Understand

A fusion category is called transparent if the associator involving any invertible object is the identity map.

• For the Haagerup-Izumi fusion rings with $G = \mathbb{Z}_{2n+1}$ (the $\mathbb{Z}_3$ case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent $F$-symbols from order $\mathcal{O}(n^6)$ to $\mathcal{O}(n^2)$, rendering the pentagon identity practically solvable.
• Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to $G = \mathbb{Z}_9$, recovering all known Haagerup-Izumi fusion categories to this order, and producing new ones.
• Transparent Haagerup-Izumi fusion categories additionally satisfying $S_4$ tetrahedral invariance are further classified up to $G = \mathbb{Z}_{15}$, and the explicit $F$-symbols for the unitary ones, including the Haagerup $\mathcal{H}_3$ fusion category, are compactly presented.